Focusing on growth and decay processes, interacting populations, and heating/cooling problems, Mathematical Modelling with Case Studies: A Differential Equations Approach using Maple(t) and MATLAB(R), Second Edition presents mathematical techniques applicable to models involving differential equations that describe rates of change. Although the authors concentrate on models involving differential equations, the ideas used can be applied to many other areas. The book carefully details the process of constructing a model, including the conversion of a seemingly complex problem into a much simpler one. It uses flow diagrams and word equations to aid in the model building process and to develop the mathematical equations. Employing theoretical, graphical, and computational tools, the authors analyze the behavior of the models under changing conditions. They discuss the validation of the models and suggest extensions to the models with an emphasis on recognizing the strengths and limitations of each model. Through applications and the tools of Maple(t) and MATLAB(R), this textbook provides hands-on model building skills.
It develops, extends, and improves simple models as well as interprets the results.
Table of Contents:
Introduction to Mathematical Modeling
Mathematical models
An overview of the book
Some modeling approaches
Modeling for decision making
Compartmental Models
Introduction
Exponential decay and radioactivity
Case study: detecting art forgeries
Case study: Pacific rats colonize New Zealand
Lake pollution models
Case study: Lake Burley Griffin
Drug assimilation into the blood
Case study: dull, dizzy, or dead?
Cascades of compartments
First-order linear DEs
Equilibrium points and stability
Case study: money, money, money makes the world go around
Models of Single Populations
Exponential growth
Density-dependent growth
Limited growth with harvesting
Case study: anchovy wipe-out
Case study: how can 2 × 106 birds mean rare?
Discrete population growth and chaos
Time-delayed regulation
Case study: Australian blowflies
Numerical Solution of Differential Equations
Introduction
Basic numerical schemes
Computer implementation using Maple and MATLAB
Instability
Discussion
Interacting Population Models
Introduction
An epidemic model for influenza
Predators and prey
Case study: Nile Perch catastrophe
Competing species
Case study: aggressive protection of lerps and nymphs
Model of a battle
Case study: rise and fall of civilizations
Phase-Plane Analysis
Introduction
Phase-plane analysis of epidemic model
Analysis of a battle model
Analysis of a predator-prey model
Analysis of competing species models
The predator-prey model revisited
Case study: bacteria battle in the gut
Linearization Analysis
Introduction
Linear theory
Applications of linear theory
Nonlinear theory
Applications of nonlinear theory
Some Extended Population Models
Introduction
Case study: competition, predation, and diversity
Extended predator-prey model
Case study: lemming mass suicides?
Case study: prickly pear meets its moth
Case study: geese defy mathematical convention
Case study: possums threaten New Zealand cows
Formulating Basic Heat Models
Introduction
Some basic physical laws
Model for a hot water heater
Heat conduction and Fourier’s law
Heat conduction through a wall
Radial heat conduction
Heat fins
Solving Time-Dependent Heat Problems
The cooling coffee problem revisited
The water heater problem revisited
Case study: it’s hot and stuffy in the attic
Spontaneous combustion
Case study: fish and chips explode
Solving Heat Conduction Problems
Boundary condition problems
Heat loss through a wall
Case study: double glazing: what’s it worth?
Insulating a water pipe
Cooling a computer chip
Introduction to Partial Differential Equations
The heat conduction equation
Oscillating soil temperatures
Case study: detecting land mines
Lake pollution revisited
Appendix A: Differential Equations
Properties of differential equations
Solution by inspection
First-order separable equations
First-order linear equations
Homogeneous equations
Inhomogeneous equations
Appendix B: Further Mathematics
Linear algebra
Partial derivatives and Taylor expansions
Review of complex numbers
Hyperbolic functions
Integration using partial fractions
Appendix C: Notes on Maple and MATLAB
Brief introduction to Maple
Using Maple to solve DEs
Brief introduction to MATLAB
Appendix D: Units and Scaling
Scaling differential equations
SI Units
References
Index
Exercises appear at the end of each chapter.
About the Author :
University of Technology, Brisbane, Australia Australian National University, Canberra, Australia
Review :
"The book is written in a very lucid manner, with numerous case studies and examples thoroughly discussed. The material is very well organized, generously illustrated, and delightfully presented. All chapters, except the first one, conclude with scores of nicely designed exercises that can be used for independent study. The book contains enough material to organize a new well-structured one-semester course or to complement the existing one with additional examples and problems and is highly recommended for either purpose" --Zentralblatt MATH, 1168 "! The book can be useful for students of mathematical modeling. They will find many skills for modeling and solving real problems. Useful sheets for Maple and MATLAB are included for numerical solution. The most important feature of the book is that it contains many real-life examples. ! The main examples are solved in detail and the others are left for the reader. This is the best treasury of real case problems seen in a single book." --EMS Newsletter, September 2009
